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arXiv · 1806.00038

Residually finite-dimensional operator algebras

Raphaël Clouâtre·Christopher Ramsey
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Abstract

We study non-selfadjoint operator algebras that can be entirely understood via their finite-dimensional representations. In contrast with the elementary matricial description of finite-dimensional $\mathrm{C}^*$-algebras, in the non-selfadjoint setting we show that an additional level of flexibility must be allowed. Motivated by this peculiarity, we consider a natural non-selfadjoint notion of residual finite-dimensionality. We identify sufficient conditions for the tensor algebra of a $\mathrm{C}^*$-correspondence to enjoy this property. To clarify the connection with the usual self-adjoint notion, we investigate the residual finite-dimensionality of the minimal and maximal $\mathrm{C}^*$-covers associated to an operator algebra.

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Raphaël Clouâtre, Christopher Ramsey. 2018-05-31. Residually finite-dimensional operator algebras. https://arxiv.org/abs/1806.00038

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